When a natural intervertebral disk has suffered significant degeneration, it is necessary to remove it and replace it with a mechanical prosthesis. As a general rule, such mechanical prostheses are made up of two elements, each element having an anchor face for anchoring in the vertebral plate of a vertebra and a co-operation face, where the co-operation faces respectively define a concave spherical cap portion and a convex spherical cap portion. The assembly thus constitutes a ball joint system that conserves for the patient the ability to perform relative movements between the two vertebrae between which the prosthesis is installed. The top and bottom elements making up the prosthesis may be single pieces, or each of them may be made up of a first part forming a prosthetic plate that is to come into contact with the vertebral plate, which part is associated with an insert that defines the concave or convex spherical cap portions.